An Optimal Convergence Theorem for Mean Curvature Flow of Arbitrary Codimension in Hyperbolic Spaces

Abstract

In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge 6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite time. In particular, we obtain the optimal differentiable sphere theorem for submanifolds in hyperbolic spaces. It should be emphasized that our pinching condition implies that the Ricci curvature of the initial submanifold is positive, but does not imply positivity of the sectional curvature of $M_0$.

24 pages

References in corpus (3)

• Positively curved surfaces in the three-sphere
• Mean curvature flow of higher codimension in Riemannian manifolds
• The extension and convergence of mean curvature flow in higher codimension

Cited by in corpus (5)

• New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory
• A New Version of Huisken's Convergence Theorem for Mean Curvature Flow in Spheres
• Ancient Solution of Mean Curvature Flow in Space Forms
• Surfaces pinched by normal curvature for mean curvature flow in space forms
• A sharp convergence theorem for the mean curvature flow in spheres I